Prime numbers are interesting in this regard not for the notional reason (low Kolmogorov complexity) but because if their high “randomness.” They are not patterned so much as they are the leftovers excluded by patterns (notably, nontrivial multiplications) but, for this reason, of interest to number theorists simply because facile pattern “shouldn’t be” there. Which I guess gets to the heart of why diagonal arguments so easily torpedo discussions of “interestingness” or the lack thereof.